Cantor Confusion
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From: Andy Smith on 2 Feb 2007 12:00 Andy Smith <Andy(a)phoenixsystems.co.uk> writes >> >>Are you familiar with Gabriel's Horn? That's the curve y = log x for x >>>= 1, rotated about the x-axis. It has finite volume and infinite >>surface area, but it is not a fractal. >> >No I wasn't, but that's quite fun. (I guess you meant to write 0<x<=1 >for the defining curve) No, that doesn't work either. I think you meant the surface of revolution of y=1/x for x>=1. Nice trumpet! -- Andy Smith
From: Dave Seaman on 2 Feb 2007 12:15 On Fri, 02 Feb 2007 16:10:59 GMT, Andy Smith wrote: > Dave Seaman <dseaman(a)no.such.host> writes >>On Fri, 02 Feb 2007 14:37:48 GMT, Andy Smith wrote: >>> Andy Smith <Andy(a)phoenixsystems.co.uk> writes >> >>> infinite volume from a finite surface area >> >>> should of course have read >> >>> infinite surface area from a finite volume >> >>Are you familiar with Gabriel's Horn? That's the curve y = log x for x >>>= 1, rotated about the x-axis. It has finite volume and infinite >>surface area, but it is not a fractal. >> > No I wasn't, but that's quite fun. (I guess you meant to write 0<x<=1 > for the defining curve) No, actually I meant y = 1/x for 1 <= x < oo. Log x is the antiderivative. -- Dave Seaman U.S. Court of Appeals to review three issues concerning case of Mumia Abu-Jamal. <http://www.mumia2000.org/>
From: David Marcus on 2 Feb 2007 13:32 Andy Smith wrote: > David Marcus <DavidMarcus(a)alumdotmit.edu> writes > >Andy Smith wrote: > (snip) > > > >> I am exercised by the idea of space filling curves, which is the same > >> issue, covering something by something of a lower classical dimension, > >> and so maybe the recursive fractal definition of reals does cover my > >> concept of line. > > > >No. This is a different problem. Even Euclid treated lines as consisting > >of points and planes as consisting of lines. A space-filling curve is a > >is a continuous function from [0,1] to [0,1]^2 that is onto. The > >surprising feature is that the function is continuous. > > > >> I asked some time before on the other thread, and didn't really get a > >> straight answer from DM, > > > >If you don't get a "straight answer", maybe you didn't ask a straight > >question. Your question was about changing the space-filling curve > >construction, but it wasn't clear how you wanted to change it. > > > >> about a fractal scenario where case 1: we > >> start of with the interval 0-1. On each iteration, each interval is > >> divided in two, and we associate a real number with the mid-point of > >> each section. After an infinite number of iterations, all the reals in > >> [0,1] are covered, no problem. > > > >Do you mean that at stage n+1, we divide all the intervals from stage n > >in half? So, at each stage we have twice as many intervals as before? If > >we do this for all natural numbers n, we only get a countable number of > >intervals (all together), so the mid-points of the intervals most > >certainly do not include all real numbers in [0,1]. The only numbers you > >get are numbers with a finite binary expansion. > > OK, I understand that argument. You are saying that I can't generate a > number with an infinite decimal expansion by a finite limit process, I don't think I'm saying that because I don't know what you mean. What is a "finite limit process"? Can you give an example? > it > just converges to some subset of Q. I can't just loosely say "an > infinite number of iterations", it's not a number. Of course you can say an "infinite number of iterations". For example, we can have an iteration for each natural number. That's certainly an infinite number of iterations. > But in that case, I misunderstand how the space filling curve iteration > can cover all the points in the plane - if you take a cross-section of > the plane along a line at e.g. 45 degrees, on iteration 1 it has 2 > points covered, and then doubling each iteration. Doesn't the same > argument then apply - you can't generate something infinite from a > finite iteration? I don't know what a "finite iteration" is. The iterations are not the final curve. Let f_n be the curve at the n-th step. Then f_n does not cover the unit square. However, we can prove that for all x in [0,1], f_n(x) converges to a point in the unit square. So, we can define f(x) = lim_{n->oo} f_n(x). And, we can prove that f is continuous and its image is the unit square. Also, note that f_n is typically composed of line segments. So, its intersection with your 45 degree line may be an entire segment. -- David Marcus
From: Lester Zick on 2 Feb 2007 13:40 On Thu, 1 Feb 2007 13:24:10 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >mueckenh(a)rz.fh-augsburg.de wrote: >> On 30 Jan., 20:25, Virgil <vir...(a)comcast.net> wrote: >> > In article <1170156979.321128.184...(a)j27g2000cwj.googlegroups.com>, >> >> > > I can't tell whether the union of all lengths is finite or not. >> > >> > As the union of all lengths is obviously equivalent to the union of all >> > finite ordinals and the union of all finite ordinals is clearly not >> > finite, WM can't tell much. >> >> The union of all lengths is not the sum of all lengths, which, >> according to your standpoint might be infinite. But the union of all >> lengths is a length. You may say that it is infinite or not. If it is >> infinite, then it corresponds to an infinite natural number. If it is >> not infinite, then the set N does not exist other than as a >> potentially infinite (= finite but unbounded) set. > >It is amazing what you can write if you fail to define your terms. If we >read it quickly, it sounds vaguely like a logical argument. Of course, >it is actually nonsense. Did you have to practice for a long time to >develop this technique? David, I can readily imagine you attended a Special Ed. course for math majors at MIT. Tell us were there any actual course requirements beyond the liturgical basis for modern math or was the degree simply a pro forma social promotion. ~v~~
From: David Marcus on 2 Feb 2007 14:09
Andy Smith wrote: > In message <epvbqo$bjl$1(a)mailhub227.itcs.purdue.edu>, Dave Seaman > <dseaman(a)no.such.host> writes > >On Fri, 02 Feb 2007 10:20:00 GMT, Andy Smith wrote: > >> Dave Seaman <dseaman(a)no.such.host> writes > >>>On Thu, 01 Feb 2007 21:52:26 GMT, Andy Smith wrote: > >>>> Dave Seaman <dseaman(a)no.such.host> writes > >>>> (snip) > >>> > >>>>>> I had previously imagined that the "Continuum Hypothesis" related to > >>>>>> whether the set of reals (as 0 dimensional points, each with no > >>>>>> neighbouring point) could cover the line, with dimension 1 (Which seemed > >>>>>> like a good question). > >>> > >>>>>You are asking whether R and "the real line" are the same set. In order > >>>>>for that question to have a nontrivial answer, you must have some meaning > >>>>>in mind for "the real line" other than R. What is your definition? > >>> > >>>> Well until recently my naive view was that points and lines were > >>>> different entities, and you might as well ask how many metres in an acre > >>>> as how many points in a line. I would have said (in your terminology) > >>>> that there is clearly an injection from the reals into the line, but > >>>> that you can never cover it. But I doubt now if that is true, or if the > >>>> question is a sensible one to start with. > >>> > >>>I don't follow. Is it your claim that: > >>> > >>> (1) A line does not have any points as members at all, or > >>> (2) A line has points as members, but perhaps there aren't > >>> enough points to fill it? > >>> > >>>Your metres-in-an-acre analogy seems to suggest (1), but otherwise you > >>>seem to be saying (2). > >>> > > > >>>If (2) is your claim, then can you show me a point on the line that is > >>>not represented by a real number? If you can do this, then evidently > >>>"the line" to you must be something other than simply "the set of real > >>>numbers", as it is to most people. Can you elaborate on this? > >>> > >>> > >> Well I meant 2) - (and I can't find any location in the acre that is not > >> covered by a metre line either). And I can't answer your question . Yes, > >> of course, the moment that you specify a real number you have defined a > >> location on the line but I am suffering from dimensional anxiety, > >> particularly when you observe that any point on the line has no nearest > >> neighbour. If I was a point sized entity wanting to walk from 0 to 1 > >> without falling down a crack, how would I do it? (I know that is not a > >> particularly sensible question, just illustrating my mis-perspective). > > > >Zeno was bothered by this when he talked about an arrow in flight being > >instantaneously "at rest" at each moment. Thanks to calculus, we now > >know that the arrow is never actually "at rest" during flight. > > > > > But the positions are reversed? I would be very happy to consider > distance along a line as a continuous function x and rate of change of > position dx/dt etc. But by asserting (or, as I understand it, defining) > the line is an infinite collection of points one reintroduces the > conceptual granularity that bothered Zeno? > You just start with points and define the real line as the set of all of > them. I have an image of a line as a continuous thing of point width, > and it is trying to marry up the perception of continuity with the set > of real points that is difficult for me. Suppose at time zero you start walking in a straight line at constant speed. At any time, you are at some point. And, you passed through each point at some time. And, the time you were at a point is the same as the distance you had travelled to get to that point. And, we can measure time (and distance) using real numbers. So, if you think lines aren't made up of points, then time isn't made up of instants. -- David Marcus |